Asymptotic results for certain first-passage times and areas of renewal processes

We consider the process $\{x-N(t):t\geq 0\}$, where $x\in \mathbb {R}_+$ and $\{N(t):t\geq 0\}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(\tau (x),A(x))$ where $\tau (x)$ is the first-passage time of $\{x-N(t):t\geq 0\}$ to reach zero or a negative value, and $A(x)≔\int _0^{\tau (x)}(x-N(t))dt$ is the corresponding first-passage (positive) area swept out by the process $\{x-N(t):t\geq 0\}$. We remark that we can define the sequence $\{(\tau (n),A(n)):n\geq 1\}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $x\to \infty$ in the fashion of large (and moderate) deviations.